Real power series/Convergence/Continuous function/Fact

Let

$f(x):=\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}^n$

be a power series and suppose that there exists some $x_0\ne 0$ such that $\sum _{n=0}^{\mathrm{\infty }}{c}_n{x}_0^n$ converges.

Then there exists a positive $R$

(where $R=\mathrm{\infty }$ is allowed) such that for all $x\in ℝ$ fulfilling $|x|<R$ the series converges absolutely. On such an (open) interval of convergence, the power series $f(x)$ represents a

continuous function.

Write a proof